Found problems: 85335
2011 Federal Competition For Advanced Students, Part 2, 3
Two circles $k_1$ and $k_2$ with radii $r_1$ and $r_2$ touch each outside at point $Q$. The other endpoints of the diameters through $Q$ are $P$ on $k_1$ and $R$ on $k_2$.
We choose two points $A$ and $B$, one on each of the arcs $PQ$ of $k_1$. ($PBQA$ is a convex quadrangle.)
Further, let $C$ be the second point of intersection of the line $AQ$ with $k_2$ and let $D$ be the second point of intersection of the line $BQ$ with $k_2$.
The lines $PB$ and $RC$ intersect in $U$ and the lines $PA$ and $RD$ intersect in $V$ .
Show that there is a point $Z$ that lies on all of these lines $UV$.
Brazil L2 Finals (OBM) - geometry, 2019.3
Let $ABC$ be an acutangle triangle inscribed in a circle $\Gamma$ of center $O$. Let $D$ be the height of the
vertex $A$. Let E and F be points over $\Gamma$ such that $AE = AD = AF$. Let $P$ and $Q$ be the intersection points of the $EF $ with sides $AB$ and $AC$ respectively. Let $X$ be the second intersection point of $\Gamma$ with the circle circumscribed to the triangle $AP Q$. Show that the lines $XD$ and $AO $ meet at a point above sobre
1976 IMO Shortlist, 8
Let $P$ be a polynomial with real coefficients such that $P(x) > 0$ if $x > 0$. Prove that there exist polynomials $Q$ and $R$ with nonnegative coefficients such that $P(x) = \frac{Q(x)}{R(x)}$ if $x > 0.$
1998 Hungary-Israel Binational, 1
A player is playing the following game. In each turn he flips a coin and guesses the outcome. If his guess is correct, he gains $ 1$ point; otherwise he loses all his points. Initially the player has no points, and plays the game
until he has $ 2$ points.
(a) Find the probability $ p_{n}$ that the game ends after exactly $ n$ flips.
(b) What is the expected number of flips needed to finish the game?
1968 Bulgaria National Olympiad, Problem 2
Find all functions $ f:\mathbb R \to \mathbb R$ such that $xf(y)+yf(x)=(x+y)f(x)f(y)$ for all reals $x$ and $y$.
2019 Nigerian Senior MO Round 3, 4
A rectangular grid whose side lengths are integers greater than $1$ is given. Smaller rectangles with area equal to an odd integer and length of each side equal to an integer greater than $1$ are cut out one by one. Finally one single unit is left. Find the least possible area of the initial grid before the cuttings.
Ps. Collected [url=https://artofproblemsolving.com/community/c949611_2019_nigerian_senior_mo_round_3]here[/url]
2014 Contests, 3
Suppose that $m$ and $n$ are relatively prime positive integers with $A = \tfrac mn$, where
\[ A = \frac{2+4+6+\dots+2014}{1+3+5+\dots+2013} - \frac{1+3+5+\dots+2013}{2+4+6+\dots+2014}. \] Find $m$. In other words, find the numerator of $A$ when $A$ is written as a fraction in simplest form.
[i]Proposed by Evan Chen[/i]
2012 Canada National Olympiad, 2
For any positive integers $n$ and $k$, let $L(n,k)$ be the least common multiple of the $k$ consecutive integers $n,n+1,\ldots ,n+k-1$. Show that for any integer $b$, there exist integers $n$ and $k$ such that $L(n,k)>bL(n+1,k)$.
2014 Postal Coaching, 1
Let $p$ be a prime such that $p\mid 2a^2-1$ for some integer $a$. Show that there exist integers $b,c$ such that $p=2b^2-c^2$.
1970 IMO Longlists, 40
Let ABC be a triangle with angles $\alpha, \beta, \gamma$ commensurable with $\pi$. Starting from a point $P$ interior to the triangle, a ball reflects on the sides of $ABC$, respecting the law of reflection that the angle of incidence is equal to the angle of reflection. Prove that, supposing that the ball never reaches any of the vertices $A,B,C$, the set of all directions in which the ball will move through time is finite. In other words, its path from the moment $0$ to infinity consists of segments parallel to a finite set of lines.
2010 Indonesia TST, 4
Let $n$ be a positive integer with $n = p^{2010}q^{2010}$ for two odd primes $p$ and $q$. Show that there exist exactly $\sqrt[2010]{n}$ positive integers $x \le n$ such that $p^{2010}|x^p - 1$ and $q^{2010}|x^q - 1$.
2024 Moldova EGMO TST, 8
In the plane there are $n$ $(n\geq4)$ marked points. There are at least $n+1$ pairs of marked points such that the distance between each pair of points is $1$. Find the greatest integer $k$ such that there is a marked point that is the center of the circle with radius $1$ on which there are at least $k$ of the marked points.
1997 AMC 12/AHSME, 16
The three row sums and the three column sums of the array
\[\begin{bmatrix} 4 & 9 & 2 \\
8 & 1 & 6 \\
3 & 5 & 7 \end{bmatrix}
\]are the same. What is the least number of entries that must be altered to make all six sums different from one another?
$ \textbf{(A)}\ 1\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 3\qquad \textbf{(D)}\ 4\qquad \textbf{(E)}\ 5$
2007 Nicolae Coculescu, 3
Determine all sets of natural numbers $ A $ that have at least two elements, and satisfying the following proposition:
$$ \forall x,y\in A\quad x>y\implies \frac{x-y}{\text{gcd} (x,y)} \in A. $$
[i]Marius Perianu[/i]
1978 Romania Team Selection Test, 5
Prove that there is no square with its four vertices on four concentric circles whose radii form an arithmetic progression.
2007 Korea National Olympiad, 2
$ ABC$ is a triangle which is not isosceles. Let the circumcenter and orthocenter of $ ABC$ be $ O$, $ H$, respectively,
and the altitudes of $ ABC$ be $ AD$, $ BC$, $ CF$. Let $ K\neq A$ be the intersection of $ AD$ and circumcircle of triangle $ ABC$, $ L$ be the intersection of $ OK$ and $ BC$, $ M$ be the midpoint of $ BC$, $ P$ be the intersection of $ AM$ and the line that passes $ L$ and perpendicular to $ BC$, $ Q$ be the intersection of $ AD$ and the line that passes $ P$ and parallel to $ MH$, $ R$ be the intersection of line $ EQ$ and $ AB$, $ S$ be the intersection of $ FD$ and $ BE$.
If $ OL \equal{} KL$, then prove that two lines $ OH$ and $ RS$ are orthogonal.
2020 Thailand TST, 1
Let $ABC$ be a triangle with circumcircle $\Gamma$. Let $\omega_0$ be a circle tangent to chord $AB$ and arc $ACB$. For each $i = 1, 2$, let $\omega_i$ be a circle tangent to $AB$ at $T_i$ , to $\omega_0$ at $S_i$ , and to arc $ACB$. Suppose $\omega_1 \ne \omega_2$. Prove that there is a circle passing through $S_1, S_2, T_1$, and $T_2$, and tangent to $\Gamma$ if and only if $\angle ACB = 90^o$.
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2018 USAJMO, 3
Let $ABCD$ be a quadrilateral inscribed in circle $\omega$ with $\overline{AC} \perp \overline{BD}$. Let $E$ and $F$ be the reflections of $D$ over lines $BA$ and $BC$, respectively, and let $P$ be the intersection of lines $BD$ and $EF$. Suppose that the circumcircle of $\triangle EPD$ meets $\omega$ at $D$ and $Q$, and the circumcircle of $\triangle FPD$ meets $\omega$ at $D$ and $R$. Show that $EQ = FR$.
1992 Polish MO Finals, 1
The functions $f_0, f_1, f_2, ...$ are defined on the reals by $f_0(x) = 8$ for all $x$, $f_{n+1}(x) = \sqrt{x^2 + 6f_n(x)}$. For all $n$ solve the equation $f_n(x) = 2x$.
2007 Federal Competition For Advanced Students, Part 2, 1
Let $ M$ be the set of all polynomials $ P(x)$ with pairwise distinct integer roots, integer coefficients and all absolut values of the coefficients less than $ 2007$. Which is the highest degree among all the polynomials of the set $ M$?
1983 USAMO, 1
On a given circle, six points $A$, $B$, $C$, $D$, $E$, and $F$ are chosen at random, independently and uniformly with respect to arc length. Determine the probability that the two triangles $ABC$ and $DEF$ are disjoint, i.e., have no common points.
2009 Jozsef Wildt International Math Competition, w. 24
If $K$, $L$, $M$ denote the midpoints of the sides $AB$, $BC$, $CA$ in triangle $\triangle ABC$, then for all $P$ in the plane of triangle $\triangle ABC$, we have $$\frac{AB}{PK}+\frac{BC}{PL}+\frac{CA}{PM} \geq \frac{AB\cdot BC \cdot CA}{4\cdot PK\cdot PL\cdot PM}$$
Indonesia MO Shortlist - geometry, g1.1
$ABCD$ is a parallelogram. $g$ is a line passing $A$. Prove that the distance from $C$ to $g$ is either the sum or the difference of the distance from $B$ to $g$, and the distance from $D$ to $g$.
1996 AIME Problems, 7
Two of the squares of a $ 7\times 7$ checkerboard are painted yellow, and the rest are painted green. Two color schemes are equivalent if one can be obtained from the other by applying a rotation in the plane of the board. How many inequivalent color schemes are possible?
2008 Chile National Olympiad, 4
Three colors are available to paint the plane. If each point in the plane is assigned one of these three colors, prove that there is a segment of length $1$ whose endpoints have the same color.